Bifurcation Model Of Successions In Ecosystems

Models of the long-term ecological successions are considered. Succession process is considered as step-bystep changing of dominant association. The model of open Eigen’s hypercycle has been used for modeling of the process. Qualitative analysis for three-dimension case has been carried out, and local bifurcations have been investigated. The process of succession can be interpreted as system’s choosing a proper level of complexity (or dimension) depending on the capacity of environment (the size of ecological niche). Connections between changing a state of the system and bifurcations in phase space is shown. INTRODUCTION Processes, which take place in ecosystems, are extremely complex; their theoretical investigation should be based on abstract concepts, which describe some general properties of the systems on the global level. One of such theoretical simplifications is the concept of succession. The succession is considered as consecutive change of one ecosystem (phitocoenose, biogeocoenoe, etc.) by other in a certain area of environment. It is not a simple transformation, but a process of simplification of the structure of the ecosystem. Any state of a system can be characterised by dominate association of species (usually phototrophic), which are “amalgamated” by other satellite species by trophic relations. Such associations appear as the key elements of biogeocoenose. Stochastic models (Culver 1981; Lippe et al. 1985; Logofet 1997; Lourival 2011) are often used for description of succession processes. The transition probability is basic parameter which determinate the dynamics of the system in such models. Such approach can be useful for simulation the system dynamics of the system, but does not reflect moving forces of the process. Concerning long-term successions, many researches (Sukachev 1972; Kogan 1977; Tilman 1990; Rabotnov 1992) emphasized essential role of competition for this processes. Use of models of competition of the Volterra type (Lepš and Prach 1981; Chakrabarti et al. 1995; Chernyshenko 1995; Weis et al. 2007) for description of succession process looks as very reasonable. Additionally to competition, there is evident positive influence of previous stages for next ones. Similar relationships between elements are described by the well-known model of hypercycle (Eigen and Schuster 1979). In the same time, relations between associations during succession have not cyclic character; the hypercycle should be open (Chernyshenko 2005). The model of open hypercycle is similar, but not equivalent to Lotka-Volterra models of the competition or “predator-prey” type. The model reflects a connection between final stage of succession (and corresponding level of complexity of the system) and the size of ecological niche. In the contribution a three-dimension case of the open hypercycle model is considered; change of the ecosystem state is interpreted mathematically as a consequence of bifurcations. OPEN EIGEN’S HYPERCYCLE Let’s consider dynamics of ecosystem, which is described by the three-dimension open hypercycle model: